Problem: $f(v, w) = (v, w, \sin(v) + \cos(w))$ What is $\dfrac{\partial f}{\partial v}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $(0, 0, 0)$ (Choice B) B $(1, 0, \cos(v))$ (Choice C) C $(1, 1, \cos(v) - \sin(w))$ (Choice D) D $(0, 1, -\sin(w))$
The partial derivative of a vector valued function is component-wise partial differentiation. $\begin{aligned} &f(v, w) = (f_0(v, w), f_1(v, w), f_2(v, w)) \\ \\ &f_v = \left( \dfrac{\partial f_0}{\partial v}, \dfrac{\partial f_1}{\partial v}, \dfrac{\partial f_2}{\partial v} \right) \\ \\ &f_w = \left( \dfrac{\partial f_0}{\partial w}, \dfrac{\partial f_1}{\partial w}, \dfrac{\partial f_2}{\partial w} \right) \end{aligned}$ Because we're taking a partial derivative with respect to $v$, we'll treat $w$ as if it were a constant. Therefore, $f_v = (1, 0, \cos(v))$.